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Description

A knot-theoretic ternary quasigroup is an algebraic system which equips a ternary operation coming from oriented (surface-)knot diagrams with region labelings. A shadow biquandle is an algebraic system which equips two binary operations and an action coming from oriented (surface-)knot diagrams with semi-arc (or semi-sheet) labelings and region labelings. Note that the region labeling by a shadow biquandle depends on the semi-arc (or semi-sheet) labeling whereas the region labeling by a knot-theoretic ternary quasigroup does not. In this talk, we show that under some condition, knot-theoretic ternary quasigroup theory and shadow biquandle theory are the same: Homology groups are the same; cocycle invariants for oriented surface-knots are the same.